Variables de décision : Pour une type de centrale $X \in [A,B,C]$, et une heure de la journée $t\in {1,\dots,24}$, on définit : $$ N_t^{(X)} = \text{Nombre de centrales } X \text{allumées à } t\text{ h, (ENTIER)} $$ $$ P_t^{(X)} = \text{Puissance totale produite par les centrales } X \text{ à } t\text{ h, (CONTINUE)} $$ Contraintes : $$ N_t^{(X)} P_{min}^{(X)} \leq P_t^{(X)} \leq N_t^{(X)} P_{max}^{(X)} \text{ , Contraintes sur la puissance totale de chaque centrale} $$ $$ 0 \leq N_t^{(X)} \leq N^{(X)} \text{ , Contraintes sur le nombre de centrales allumées possible} $$ $$ \forall t, \sum_X P_t^{(X)} = D_t \text{ , Contrainte équilibre offre-demande} $$ Objectif : $$ \text{Minimiser} \sum_X \sum_T P_t^{(X)} C_{MWh}^{(X)} $$
Set parameter TokenServer to value "dev.cma.mines-paristech.fr"
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 168 rows, 144 columns and 360 nonzeros
Model fingerprint: 0xf31da896
Variable types: 72 continuous, 72 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 4e+03]
Objective range [1e+00, 3e+00]
Bounds range [5e+00, 1e+01]
RHS range [2e+04, 4e+04]
Found heuristic solution: objective 1235375.0000
Presolve removed 162 rows and 139 columns
Presolve time: 0.02s
Presolved: 6 rows, 5 columns, 14 nonzeros
Found heuristic solution: objective 881275.00000
Variable types: 2 continuous, 3 integer (0 binary)
Root relaxation: objective 8.694000e+05, 2 iterations, 0.00 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 869400.000 0 1 881275.000 869400.000 1.35% - 0s
H 0 0 869400.00000 869400.000 0.00% - 0s
0 0 869400.000 0 1 869400.000 869400.000 0.00% - 0s
Explored 1 nodes (2 simplex iterations) in 0.04 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)
Solution count 3: 869400 881275 1.23538e+06
Optimal solution found (tolerance 1.00e-04)
Best objective 8.694000000000e+05, best bound 8.694000000000e+05, gap 0.0000%
Coût : 869400.0
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 168 rows, 144 columns and 360 nonzeros
Model fingerprint: 0x635b27b9
Coefficient statistics:
Matrix range [1e+00, 4e+03]
Objective range [1e+00, 3e+00]
Bounds range [5e+00, 1e+01]
RHS range [2e+04, 4e+04]
Presolve removed 153 rows and 99 columns
Presolve time: 0.01s
Presolved: 15 rows, 45 columns, 45 nonzeros
Iteration Objective Primal Inf. Dual Inf. Time
0 8.5266000e+05 1.743750e+04 0.000000e+00 0s
15 8.6940000e+05 0.000000e+00 0.000000e+00 0s
Solved in 15 iterations and 0.01 seconds (0.00 work units)
Optimal objective 8.694000000e+05
Coût : 869400.0
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 168 rows, 144 columns and 360 nonzeros
Model fingerprint: 0x7066cec8
Variable types: 72 continuous, 72 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 4e+03]
Objective range [1e+00, 2e+03]
Bounds range [5e+00, 1e+01]
RHS range [2e+04, 4e+04]
Found heuristic solution: objective 1271875.0000
Presolve removed 162 rows and 139 columns
Presolve time: 0.02s
Presolved: 6 rows, 5 columns, 14 nonzeros
Found heuristic solution: objective 987500.00000
Variable types: 2 continuous, 3 integer (0 binary)
Found heuristic solution: objective 985950.00000
Root relaxation: objective 9.787500e+05, 2 iterations, 0.00 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 978750.000 0 1 985950.000 978750.000 0.73% - 0s
H 0 0 978900.00000 978750.000 0.02% - 0s
0 0 978750.000 0 1 978900.000 978750.000 0.02% - 0s
Explored 1 nodes (2 simplex iterations) in 0.04 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)
Solution count 4: 978900 985950 987500 1.27188e+06
Optimal solution found (tolerance 1.00e-04)
Best objective 9.789000000000e+05, best bound 9.789000000000e+05, gap 0.0000%
Coût : 978900.0
Pour ce problème on modifie les variables de décisions et on y adapte les contraintes.
Variables de décision : Pour une type de centrale $X \in [A,B,C]$, et une heure de la journée $t\in {1,\dots,24}$, on définit : $$ N_t^{(X)} = \text{Nombre de centrales } X \text{allumées à } t\text{ h, (ENTIER)} $$ $$ N_{start,t}^{(X)} = \text{Nombre de centrales } X \text{démarrées à } t\text{ h, (ENTIER)} $$ $$ P_t^{(X)} = \text{Puissance totale produite par les centrales } X \text{ à } t\text{ h, (CONTINUE)} $$ Contraintes (par convention $N_{-1}^{(X)}=0$): $$ N_t^{(X)} P_{min}^{(X)} \leq P_t^{(X)} \leq N_t^{(X)} P_{max}^{(X)} \text{ , Contraintes sur la puissance totale de chaque centrale} $$
Objectif : $$ \text{Minimiser} \sum_X \sum_T (P_t^{(X)} - P_{min}^{(X)}N_t^{(X)}) C_{MWh}^{(X)} + N_t^{(X)}C_{base}^{(X)} + N_{start,t}^{(X)}C_{start}^{(X)} $$
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 312 rows, 216 columns and 714 nonzeros
Model fingerprint: 0xafd71f6e
Variable types: 72 continuous, 144 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 4e+03]
Objective range [1e+00, 2e+03]
Bounds range [5e+00, 1e+01]
RHS range [5e+00, 4e+04]
Found heuristic solution: objective 1474375.0000
Presolve removed 108 rows and 30 columns
Presolve time: 0.00s
Presolved: 204 rows, 186 columns, 522 nonzeros
Variable types: 42 continuous, 144 integer (0 binary)
Found heuristic solution: objective 1378930.0000
Root relaxation: objective 1.011257e+06, 52 iterations, 0.00 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 1011257.14 0 26 1378930.00 1011257.14 26.7% - 0s
H 0 0 1019660.0000 1011257.14 0.82% - 0s
H 0 0 1018090.0000 1011257.14 0.67% - 0s
H 0 0 1016340.0000 1011257.14 0.50% - 0s
H 0 0 1016100.0000 1011257.14 0.48% - 0s
0 0 1014233.33 0 4 1016100.00 1014233.33 0.18% - 0s
H 0 0 1015860.0000 1014233.33 0.16% - 0s
H 0 0 1014650.0000 1014233.33 0.04% - 0s
H 0 0 1014400.0000 1014233.33 0.02% - 0s
Cutting planes:
Gomory: 3
MIR: 27
StrongCG: 2
Explored 1 nodes (92 simplex iterations) in 0.03 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)
Solution count 9: 1.0144e+06 1.01465e+06 1.01586e+06 ... 1.47438e+06
Optimal solution found (tolerance 1.00e-04)
Best objective 1.014400000000e+06, best bound 1.014400000000e+06, gap 0.0000%
Pour intégrer la réserve de puissance on ajoute la contrainte : $$ \sum_X N_t^{(X)}P_{max}^{(X)}\geq D_t\times 1,15 $$
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 336 rows, 216 columns and 786 nonzeros
Model fingerprint: 0x11b5f8ee
Variable types: 72 continuous, 144 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 4e+03]
Objective range [1e+00, 2e+03]
Bounds range [5e+00, 1e+01]
RHS range [5e+00, 5e+04]
MIP start from previous solve did not produce a new incumbent solution
MIP start from previous solve violates constraint Réserve_de_puissance_à_15 by 4500.000000000
Found heuristic solution: objective 1316565.0000
Presolve removed 108 rows and 30 columns
Presolve time: 0.00s
Presolved: 228 rows, 186 columns, 594 nonzeros
Found heuristic solution: objective 1229165.0000
Variable types: 42 continuous, 144 integer (0 binary)
Found heuristic solution: objective 1198245.0000
Root relaxation: objective 1.012257e+06, 59 iterations, 0.00 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 1012257.14 0 26 1198245.00 1012257.14 15.5% - 0s
H 0 0 1020660.0000 1012257.14 0.82% - 0s
H 0 0 1017340.0000 1012257.14 0.50% - 0s
H 0 0 1017100.0000 1012257.14 0.48% - 0s
0 0 1015150.00 0 12 1017100.00 1015150.00 0.19% - 0s
H 0 0 1016860.0000 1015150.00 0.17% - 0s
H 0 0 1015725.0000 1015150.00 0.06% - 0s
H 0 0 1015150.0000 1015150.00 0.00% - 0s
Cutting planes:
Gomory: 1
MIR: 25
Explored 1 nodes (94 simplex iterations) in 0.04 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)
Solution count 9: 1.01515e+06 1.01572e+06 1.01686e+06 ... 1.31657e+06
Optimal solution found (tolerance 1.00e-04)
Best objective 1.015150000000e+06, best bound 1.015150000000e+06, gap 0.0000%
Pour ce problème on modifie la convention $N_{-1}^{(X)}=0$ par $N_{-1}^{(X)}=N_{23}^{(X)}$
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 336 rows, 216 columns and 792 nonzeros
Model fingerprint: 0xfaf92924
Variable types: 72 continuous, 144 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 4e+03]
Objective range [1e+00, 2e+03]
Bounds range [5e+00, 1e+01]
RHS range [5e+00, 5e+04]
Found heuristic solution: objective 1313700.0000
Presolve removed 105 rows and 27 columns
Presolve time: 0.00s
Presolved: 231 rows, 189 columns, 603 nonzeros
Found heuristic solution: objective 1225165.0000
Variable types: 42 continuous, 147 integer (0 binary)
Found heuristic solution: objective 1192195.0000
Root relaxation: objective 9.855143e+05, 54 iterations, 0.00 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 985514.286 0 26 1192195.00 985514.286 17.3% - 0s
H 0 0 993725.00000 985514.286 0.83% - 0s
H 0 0 990405.00000 985514.286 0.49% - 0s
H 0 0 990165.00000 985514.286 0.47% - 0s
H 0 0 989030.00000 988540.000 0.05% - 0s
0 0 988540.000 0 3 989030.000 988540.000 0.05% - 0s
H 0 0 988540.00000 988540.000 0.00% - 0s
0 0 988540.000 0 3 988540.000 988540.000 0.00% - 0s
Cutting planes:
MIR: 18
Explored 1 nodes (85 simplex iterations) in 0.03 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)
Solution count 8: 988540 989030 990165 ... 1.3137e+06
Optimal solution found (tolerance 1.00e-04)
Best objective 9.885400000000e+05, best bound 9.885400000000e+05, gap 0.0000%
On ajoute les variables de décisions suivantes:
pour $Y\in[9,14]$ (pour les centrales de 900 MW et 1400MW),
$$
H_t^{(Y)} \in \{0,1\} \text{ , vaut 1 si la centrale $Y$ fonctionne à } t \text{ h 0 sinon}
$$
$$
H_{start,t}^{(Y)} \in \{0,1\} \text{ , vaut 1 si la centrale $Y$ démarre à } t \text{ h 0 sinon}
$$
Avec les contraintes :
$$
H_{t}^{(Y)} \leq H_{start,t}^{(Y)} + H_{t-1}^{(Y)} \text{, (5.1.1) Si la centrale fonctionne alors elle était allumée ou elle démarre}
$$
$$
\sum_X P_t^{(X)} + \sum_Y N_t^{(Y)}P^{(Y)}= D_t \text{, (5.1.2) Contrainte équilibre offre-demande}
$$
$$
\sum_X N_t^{(X)}P_{max}^{(X)} + \sum_Y P^{(Y)}\geq D_t\times 1,15 \text{, (5.1.3) marges de sécurité}
$$
L'objectif mis à jour devient:
$$
\text{Minimiser} \sum_X \sum_t (P_t^{(X)} - P_{min}^{(X)}N_t^{(X)}) C_{MWh}^{(X)} + N_t^{(X)}C_{base}^{(X)} + N_{start,t}^{(X)}C_{start}^{(X)} + \sum_Y \sum_t H_t^{(Y)}C_{base}^{(Y)} + H_t^{(Y)}C_{start}^{(Y)}
$$
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 384 rows, 312 columns and 984 nonzeros
Model fingerprint: 0x62d80b5b
Variable types: 72 continuous, 240 integer (96 binary)
Coefficient statistics:
Matrix range [1e+00, 4e+03]
Objective range [1e+00, 2e+03]
Bounds range [1e+00, 1e+01]
RHS range [5e+00, 4e+04]
Found heuristic solution: objective 1326520.0000
Presolve removed 72 rows and 0 columns
Presolve time: 0.00s
Presolved: 312 rows, 312 columns, 840 nonzeros
Variable types: 72 continuous, 240 integer (96 binary)
Found heuristic solution: objective 1308745.0000
Root relaxation: objective 8.884163e+05, 90 iterations, 0.00 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 888416.250 0 30 1308745.00 888416.250 32.1% - 0s
H 0 0 906941.25000 888416.250 2.04% - 0s
H 0 0 893135.00000 888416.250 0.53% - 0s
H 0 0 890510.00000 888416.250 0.24% - 0s
H 0 0 890260.00000 888474.286 0.20% - 0s
0 0 890176.667 0 22 890260.000 890176.667 0.01% - 0s
Cutting planes:
Gomory: 2
MIR: 33
Explored 1 nodes (132 simplex iterations) in 0.03 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)
Solution count 6: 890260 890510 893135 ... 1.32652e+06
Optimal solution found (tolerance 1.00e-04)
Best objective 8.902600000000e+05, best bound 8.901766666667e+05, gap 0.0094%
On ajoute les variables de décision suivante :
$$
S_t = \text{Puissance appelée par le pompage à l'instant }t
$$
Puis la contrainte :
$$
\sum_tS_t d^{(S)} = \sum_{t,Y} H_t^{(Y)}d^{(Y)} \text{ , L'eau extraite doit être réinjectée sur un cycle d'une journée}
$$
Où $d^{(S)}$ représente la hauteur d'eau élevée par MWh et $d^{(Y)}$ la hauteur d'eau prélevée lorsque la centrale $Y$ fonctionne.
On met à jour les contraintes suivante:
$$
\sum_X P_t^{(X)} + \sum_Y H_t^{(Y)}P^{(Y)}-S_t= D_t \text{ , Contrainte équilibre offre-demande}
$$
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 385 rows, 336 columns and 1080 nonzeros
Model fingerprint: 0x2a90ce11
Variable types: 96 continuous, 240 integer (96 binary)
Coefficient statistics:
Matrix range [3e-04, 4e+03]
Objective range [1e+00, 2e+03]
Bounds range [1e+00, 1e+01]
RHS range [5e+00, 4e+04]
Found heuristic solution: objective 1491600.0000
Presolve removed 72 rows and 0 columns
Presolve time: 0.00s
Presolved: 313 rows, 336 columns, 936 nonzeros
Variable types: 96 continuous, 240 integer (96 binary)
Found heuristic solution: objective 1463225.0000
Root relaxation: objective 9.850143e+05, 145 iterations, 0.00 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 985014.286 0 26 1463225.00 985014.286 32.7% - 0s
H 0 0 994935.00000 985014.286 1.00% - 0s
H 0 0 991890.00000 985014.286 0.69% - 0s
H 0 0 990145.00000 985014.286 0.52% - 0s
0 0 985814.126 0 48 990145.000 985814.126 0.44% - 0s
0 0 985814.126 0 48 990145.000 985814.126 0.44% - 0s
H 0 0 988290.00000 985814.126 0.25% - 0s
H 0 0 988040.00000 985814.126 0.23% - 0s
0 2 985814.126 0 48 988040.000 985814.126 0.23% - 0s
H 2 4 988002.00000 986022.114 0.20% 4.0 0s
H 5 8 986630.00000 986022.114 0.06% 8.8 0s
H 21 13 986290.00000 986051.521 0.02% 8.0 0s
Cutting planes:
Gomory: 5
MIR: 24
Explored 45 nodes (439 simplex iterations) in 0.11 seconds (0.04 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 986290 986630 988002 ... 1.4916e+06
Optimal solution found (tolerance 1.00e-04)
Best objective 9.862900000000e+05, best bound 9.862900000000e+05, gap 0.0000%
Les variables de décisions concernant l'hydroélectricité deviennent : $$ H_t^{(Y,n)} \in \{0,1\} = \text{La centrale hydroélectrique } Y \text{ fonctionne au palier } n \text{ à l'instant } t $$ $$ H_{start,t}^{(Y)} \in \{0,1\} = \text{ La centrale hydroélectrique $Y$ démarre à l'instant $t$} $$ Les contraintes suivantes sont modifiées :
$$ \text{(5.3.1) Equilibre offre-demande : } \sum_X P_t^{(X)} + \sum_{Y,n} H_t^{(Y,n)}P^{(Y,n)}-S_t= D_t $$$$ \text{(5.3.2) Réserve de puissance : } \sum_X N_t^{(X)}P_{max}^{(X)} + \sum_Y P^{(Y,n_{max})}\geq D_t\times 1,15 $$$$ \text{(5.3.3) Niveau réservoir : } \sum_t S_t d^{(S)} = \sum_{t,Y,n} H_t^{(Y,n)}d^{(Y,n)} $$Les contraintes sur les démarrages et les paliers deviennent : $$ \text{(5.3.4) Un seul palier fonctionne à la fois , }\sum_n H_t^{(Y,n)} \leq 1 $$ $$ \text{(5.3.5) Si un palier fonctionne alors il y avait déjà un palier actif ou la centrale est démarrée , } H_t^{(Y,n)} \leq \sum_n H_{t-1}^{(Y,n)} + H_{start,t}^{(Y)} $$ Modification de la fonction objectif : $$ \text{Minimiser} \sum_X \sum_t (P_t^{(X)} - P_{min}^{(X)}N_t^{(X)}) C_{MWh}^{(X)} + N_t^{(X)}C_{base}^{(X)} + N_{start,t}^{(X)}C_{start}^{(X)} + \sum_{Y,n} \sum_t H_t^{(Y,n)}C_{base}^{(Y,n)} + \sum_Y H_{start,t}^{(Y)}C_{start}^{(Y)} $$ Où $P^{(Y,n)}$ est la puissance de la centrale $Y$ au palier $n$, $d^{(Y,n)}$ est l'abaissement sur une heure quand la centrale $Y$ fonctionne au palier $n$ et $C_{base}^{(Y,n)}$ est le coût du fonctionnement de la centrale $Y$ pendant une heure de fonctionnement au palier $n$
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 433 rows, 480 columns and 1848 nonzeros
Model fingerprint: 0xd3c4d88d
Variable types: 96 continuous, 384 integer (240 binary)
Coefficient statistics:
Matrix range [3e-04, 4e+03]
Objective range [1e+00, 2e+03]
Bounds range [1e+00, 1e+01]
RHS range [1e+00, 4e+04]
Presolve removed 72 rows and 0 columns
Presolve time: 0.00s
Presolved: 361 rows, 480 columns, 1704 nonzeros
Variable types: 96 continuous, 384 integer (240 binary)
Found heuristic solution: objective 1114770.0000
Found heuristic solution: objective 1086990.0000
Found heuristic solution: objective 1026475.0000
Root relaxation: objective 9.850143e+05, 244 iterations, 0.00 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 985014.286 0 26 1026475.00 985014.286 4.04% - 0s
H 0 0 994935.00000 985014.286 1.00% - 0s
H 0 0 990145.00000 985014.286 0.52% - 0s
H 0 0 987635.00000 985677.273 0.20% - 0s
0 0 985677.273 0 48 987635.000 985677.273 0.20% - 0s
H 0 0 987176.00000 985677.273 0.15% - 0s
0 0 985681.649 0 48 987176.000 985681.649 0.15% - 0s
0 0 985681.649 0 26 987176.000 985681.649 0.15% - 0s
0 0 985681.649 0 48 987176.000 985681.649 0.15% - 0s
H 0 0 987076.00000 985681.649 0.14% - 0s
0 0 985687.391 0 48 987076.000 985687.391 0.14% - 0s
H 0 0 986256.00000 985687.391 0.06% - 0s
0 0 985687.391 0 21 986256.000 985687.391 0.06% - 0s
0 0 985687.391 0 45 986256.000 985687.391 0.06% - 0s
0 0 985687.391 0 41 986256.000 985687.391 0.06% - 0s
0 0 985687.391 0 41 986256.000 985687.391 0.06% - 0s
0 2 985687.391 0 41 986256.000 985687.391 0.06% - 0s
H 28 7 986160.00000 985789.464 0.04% 9.4 0s
Cutting planes:
Gomory: 2
MIR: 22
Explored 50 nodes (1170 simplex iterations) in 0.21 seconds (0.07 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 986160 986256 987076 ... 1.11477e+06
Optimal solution found (tolerance 1.00e-04)
Best objective 9.861600000000e+05, best bound 9.861380000000e+05, gap 0.0022%
On introduit la variable de décision suivante : $$ N^{S}_t \in \{0,1\} = \text{La pompage est en fonctionnement} $$ On ajoute les contraintes suivante : $$ \text{(5.4.1) Si une centrale hydroélectrique est activée alors le pompage est désactivé , } \frac{1}{\text{Nb centrale hydro}}\sum_{Y,n}H_t^{(Y,n)} \leq 1 - N_t^{(S)} $$ Il est peut être plus optimal de formuler (5.4.1) de cette manière: $$ H_t^{(Y,n)} \leq 1 - N_t^{(S)} $$ $$ \text{(5.4.2) Si le pompage est allumé alors il peut y avoir du pompage , } S_t \leq M \times N_t^{(S)} $$ Où $M$ est une quantité égal à l'ensemble de la capacité installé du parc (borne sup de la valeur que peut prendre $S_t$)
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 481 rows, 504 columns and 2112 nonzeros
Model fingerprint: 0x9e89f7d2
Variable types: 96 continuous, 408 integer (264 binary)
Coefficient statistics:
Matrix range [3e-04, 6e+04]
Objective range [1e+00, 2e+03]
Bounds range [1e+00, 1e+01]
RHS range [1e+00, 4e+04]
Presolve removed 96 rows and 0 columns
Presolve time: 0.01s
Presolved: 385 rows, 504 columns, 1800 nonzeros
Variable types: 96 continuous, 408 integer (264 binary)
Found heuristic solution: objective 1155935.0000
Found heuristic solution: objective 1086990.0000
Found heuristic solution: objective 1029100.0000
Root relaxation: objective 9.850143e+05, 210 iterations, 0.00 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 985014.286 0 26 1029100.00 985014.286 4.28% - 0s
H 0 0 994935.00000 985014.286 1.00% - 0s
H 0 0 990145.00000 985014.286 0.52% - 0s
H 0 0 989730.00000 985677.273 0.41% - 0s
0 0 985677.273 0 66 989730.000 985677.273 0.41% - 0s
0 0 985692.849 0 72 989730.000 985692.849 0.41% - 0s
H 0 0 988290.00000 985692.849 0.26% - 0s
0 0 985697.389 0 72 988290.000 985697.389 0.26% - 0s
H 0 0 988040.00000 985697.389 0.24% - 0s
0 2 985697.389 0 72 988040.000 985697.389 0.24% - 0s
H 35 40 986855.00000 986007.287 0.09% 16.7 0s
H 75 56 986771.00000 986031.356 0.07% 13.3 0s
Cutting planes:
MIR: 54
Explored 335 nodes (4215 simplex iterations) in 0.20 seconds (0.14 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 986771 986771 986855 ... 1.08699e+06
Optimal solution found (tolerance 1.00e-04)
Best objective 9.867710000000e+05, best bound 9.866744586729e+05, gap 0.0098%
Variables de décision : Pour une type de centrale $X \in [A,B,C]$, $k$ l'identification de la $k$-ieme centrale, et une heure de la journée $t\in {1,\dots,24}$, on définit : $$ N_t^{(X,k)} \in\{0,1\} \text{La $k$-ieme centrale $X$ fonctionne à $t$} $$ $$ N_{start,t}^{(X,k)} \in\{0,1\} \text{La $k$-ieme centrale $X$ est démarrée à $t$} $$ $$ P_t^{(X,k)} = \text{Puissance de la $k$-ième centrale $X$ à l'instant $t$} $$ Contraintes : $$ N_t^{(X,k)} P_{min}^{(X)} \leq P_t^{(X,k)} \leq N_t^{(X,k)} P_{max}^{(X)} \text{ , (6.1.1) Contraintes sur la puissance totale de chaque centrale} $$ $$ N_t^{(X,k)} \leq N_{t-1}^{(X,k)}+N_{start,t}^{(X,k)} \text{ , (6.1.2) Contraintes sur la puissance totale de chaque centrale} $$ $$ \text{(6.1.3) Equilibre offre-demande : } \sum_X \sum_{k=1}^{N^{(X)}} P_t^{(X,k)} + \sum_{Y,n} H_t^{(Y,n)}P^{(Y,n)}-S_t= D_t $$ $$ \text{(6.1.4) Réserve de puissance : } \sum_X \sum_{k=1}^{N^{(X)}} N_t^{(X,k)}P_{max}^{(X)} + \sum_Y P^{(Y,n_{max})}\geq D_t\times 1,15 $$ Objectif : $$ \text{Minimiser} \sum_X \sum_{k=1}^{N^{(X)}} \sum_t (P_t^{(X,k)} - P_{min}^{(X)}N_t^{(X,k)}) C_{MWh}^{(X)} + N_t^{(X,k)}C_{base}^{(X)} + N_{start,t}^{(X,k)}C_{start}^{(X)} + \sum_{Y,n} \sum_t H_t^{(Y,n)}C_{base}^{(Y,n)} + \sum_Y H_{start,t}^{(Y)}C_{start}^{(Y)} $$
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2137 rows, 2232 columns and 7152 nonzeros
Model fingerprint: 0x3a60e45c
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
Matrix range [3e-04, 6e+04]
Objective range [1e+00, 2e+03]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 4e+04]
Presolve removed 24 rows and 0 columns
Presolve time: 0.01s
Presolved: 2113 rows, 2232 columns, 6984 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)
Found heuristic solution: objective 1066070.0000
Found heuristic solution: objective 1049400.0000
Found heuristic solution: objective 1031950.0000
Root relaxation: objective 9.848205e+05, 1445 iterations, 0.01 seconds (0.01 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 984820.536 0 31 1031950.00 984820.536 4.57% - 0s
H 0 0 990145.00000 984820.536 0.54% - 0s
H 0 0 989980.00000 985527.783 0.45% - 0s
0 0 985527.783 0 75 989980.000 985527.783 0.45% - 0s
H 0 0 989730.00000 985527.783 0.42% - 0s
0 0 985527.783 0 75 989730.000 985527.783 0.42% - 0s
0 0 985550.204 0 98 989730.000 985550.204 0.42% - 0s
0 0 985550.204 0 98 989730.000 985550.204 0.42% - 0s
H 0 0 988837.00000 985550.204 0.33% - 0s
0 0 985680.000 0 100 988837.000 985680.000 0.32% - 0s
0 0 985680.000 0 105 988837.000 985680.000 0.32% - 0s
H 0 0 988217.00000 985680.000 0.26% - 0s
0 0 985680.000 0 94 988217.000 985680.000 0.26% - 0s
0 0 985680.000 0 73 988217.000 985680.000 0.26% - 0s
0 0 985680.000 0 73 988217.000 985680.000 0.26% - 0s
0 0 985680.000 0 73 988217.000 985680.000 0.26% - 0s
0 2 985680.000 0 73 988217.000 985680.000 0.26% - 0s
H 71 76 987962.00000 985990.728 0.20% 22.9 0s
H 168 157 987850.00000 985990.728 0.19% 16.6 0s
H 210 206 987390.00000 986017.559 0.14% 15.9 0s
H 347 236 986818.00000 986017.559 0.08% 13.2 1s
10174 1086 986623.835 40 19 986818.000 986623.835 0.02% 8.4 5s
H15465 639 986771.00000 986657.979 0.01% 8.5 5s
Cutting planes:
Lift-and-project: 5
Cover: 2
Projected implied bound: 7
MIR: 22
Flow cover: 9
Inf proof: 1
Relax-and-lift: 1
Explored 16562 nodes (141877 simplex iterations) in 6.11 seconds (6.30 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 986771 986818 987390 ... 990145
Optimal solution found (tolerance 1.00e-04)
Best objective 9.867710000000e+05, best bound 9.867148571429e+05, gap 0.0057%
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2137 rows, 2232 columns and 7152 nonzeros
Model fingerprint: 0x2d7f75ce
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
Matrix range [3e-04, 6e+04]
Objective range [1e+00, 2e+03]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 5e+04]
Presolve removed 24 rows and 0 columns
Presolve time: 0.01s
Presolved: 2113 rows, 2232 columns, 6984 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)
Found heuristic solution: objective 1070205.0000
Found heuristic solution: objective 1068650.0000
Root relaxation: objective 9.865919e+05, 1422 iterations, 0.01 seconds (0.01 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 986591.867 0 57 1068650.00 986591.867 7.68% - 0s
H 0 0 1001855.0000 986591.867 1.52% - 0s
H 0 0 999480.00000 986591.867 1.29% - 0s
H 0 0 992900.00000 986591.867 0.64% - 0s
0 0 986628.571 0 100 992900.000 986628.571 0.63% - 0s
0 0 986659.053 0 93 992900.000 986659.053 0.63% - 0s
0 0 986748.434 0 91 992900.000 986748.434 0.62% - 0s
0 0 986757.183 0 100 992900.000 986757.183 0.62% - 0s
0 0 986757.183 0 109 992900.000 986757.183 0.62% - 0s
H 0 0 990449.00000 986757.183 0.37% - 0s
0 0 986805.247 0 102 990449.000 986805.247 0.37% - 0s
0 0 986822.410 0 79 990449.000 986822.410 0.37% - 0s
0 0 986824.328 0 89 990449.000 986824.328 0.37% - 0s
H 0 0 989002.00000 986824.328 0.22% - 0s
0 0 986824.328 0 114 989002.000 986824.328 0.22% - 0s
H 0 0 988370.00000 986824.328 0.16% - 0s
0 0 986828.816 0 128 988370.000 986828.816 0.16% - 0s
0 0 986828.816 0 58 988370.000 986828.816 0.16% - 0s
0 0 986828.816 0 124 988370.000 986828.816 0.16% - 0s
0 0 986828.816 0 130 988370.000 986828.816 0.16% - 0s
0 0 986828.816 0 123 988370.000 986828.816 0.16% - 0s
0 0 986828.816 0 126 988370.000 986828.816 0.16% - 0s
0 0 986860.182 0 180 988370.000 986860.182 0.15% - 0s
0 0 986882.215 0 171 988370.000 986882.215 0.15% - 0s
0 0 986882.300 0 133 988370.000 986882.300 0.15% - 0s
0 0 986909.848 0 181 988370.000 986909.848 0.15% - 0s
0 0 986910.326 0 135 988370.000 986910.326 0.15% - 0s
0 2 986910.326 0 93 988370.000 986910.326 0.15% - 0s
H 33 39 988150.00000 987203.586 0.10% 29.2 0s
H 116 119 988115.00000 987203.586 0.09% 16.0 0s
H 172 151 987865.00000 987222.801 0.07% 13.8 0s
H 246 205 987762.00000 987222.801 0.05% 13.0 0s
H 479 247 987641.00000 987310.213 0.03% 10.7 1s
* 697 343 36 987631.00000 987310.416 0.03% 9.5 1s
H 1951 815 987620.00000 987316.696 0.03% 8.3 1s
H 2410 996 987615.00000 987319.309 0.03% 8.0 1s
16862 5776 cutoff 62 987615.000 987480.331 0.01% 7.6 5s
30040 10674 987611.429 54 133 987615.000 987485.387 0.01% 7.0 10s
Cutting planes:
Gomory: 1
Cover: 1
MIR: 9
StrongCG: 1
Flow cover: 8
Explored 30099 nodes (217021 simplex iterations) in 12.20 seconds (7.82 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 987615 987615 987615 ... 987641
Optimal solution found (tolerance 1.00e-04)
Best objective 9.876150000000e+05, best bound 9.875226039275e+05, gap 0.0094%
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2137 rows, 2232 columns and 7152 nonzeros
Model fingerprint: 0x11a09cba
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
Matrix range [3e-04, 6e+04]
Objective range [1e+00, 2e+03]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 4e+04]
Presolve removed 24 rows and 0 columns
Presolve time: 0.01s
Presolved: 2113 rows, 2232 columns, 6984 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)
Found heuristic solution: objective 1065810.0000
Found heuristic solution: objective 1049600.0000
Root relaxation: objective 9.892817e+05, 1484 iterations, 0.01 seconds (0.02 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 989281.715 0 54 1049600.00 989281.715 5.75% - 0s
H 0 0 1008655.0000 989281.715 1.92% - 0s
H 0 0 1003320.0000 989281.715 1.40% - 0s
H 0 0 991750.00000 989281.715 0.25% - 0s
0 0 989568.409 0 49 991750.000 989568.409 0.22% - 0s
0 0 989568.409 0 49 991750.000 989568.409 0.22% - 0s
H 0 0 990860.00000 989568.409 0.13% - 0s
0 0 989583.635 0 54 990860.000 989583.635 0.13% - 0s
0 0 989583.635 0 54 990860.000 989583.635 0.13% - 0s
H 0 0 990500.00000 989583.635 0.09% - 0s
0 0 989583.635 0 60 990500.000 989583.635 0.09% - 0s
0 0 989594.782 0 68 990500.000 989594.782 0.09% - 0s
0 0 989609.740 0 79 990500.000 989609.740 0.09% - 0s
0 0 989609.978 0 81 990500.000 989609.978 0.09% - 0s
0 0 989733.996 0 86 990500.000 989733.996 0.08% - 0s
0 0 989733.996 0 60 990500.000 989733.996 0.08% - 0s
0 0 989734.282 0 92 990500.000 989734.282 0.08% - 0s
0 0 989734.807 0 95 990500.000 989734.807 0.08% - 0s
0 0 989734.807 0 95 990500.000 989734.807 0.08% - 0s
0 0 989734.807 0 101 990500.000 989734.807 0.08% - 0s
0 0 989734.807 0 93 990500.000 989734.807 0.08% - 0s
H 0 0 990440.00000 989734.807 0.07% - 0s
0 0 989734.807 0 42 990440.000 989734.807 0.07% - 0s
0 0 989734.807 0 47 990440.000 989734.807 0.07% - 0s
0 0 989757.223 0 43 990440.000 989757.223 0.07% - 0s
0 0 989757.714 0 56 990440.000 989757.714 0.07% - 0s
0 0 989757.714 0 76 990440.000 989757.714 0.07% - 0s
0 0 989757.987 0 75 990440.000 989757.987 0.07% - 0s
0 0 989758.118 0 75 990440.000 989758.118 0.07% - 0s
0 0 989758.118 0 89 990440.000 989758.118 0.07% - 0s
0 0 989758.118 0 89 990440.000 989758.118 0.07% - 0s
0 0 989763.110 0 49 990440.000 989763.110 0.07% - 0s
0 0 989763.199 0 41 990440.000 989763.199 0.07% - 0s
0 0 989778.249 0 41 990440.000 989778.249 0.07% - 0s
0 0 989897.866 0 28 990440.000 989897.866 0.05% - 0s
0 0 989897.866 0 28 990440.000 989897.866 0.05% - 0s
0 2 989897.866 0 28 990440.000 989897.866 0.05% - 0s
* 434 281 46 990430.00000 990122.540 0.03% 5.3 1s
H 979 626 990420.00000 990122.540 0.03% 5.3 1s
H 1701 719 990410.00000 990122.540 0.03% 6.1 2s
Cutting planes:
Gomory: 9
Lift-and-project: 1
MIR: 79
Flow cover: 53
RLT: 2
Explored 26619 nodes (185253 simplex iterations) in 4.99 seconds (3.82 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 990410 990420 990430 ... 991750
Optimal solution found (tolerance 1.00e-04)
Best objective 9.904100000000e+05, best bound 9.903183333333e+05, gap 0.0093%
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2150 rows, 2232 columns and 7308 nonzeros
Model fingerprint: 0xd000b483
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
Matrix range [3e-04, 6e+04]
Objective range [1e+00, 2e+03]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 4e+04]
Presolve removed 24 rows and 0 columns
Presolve time: 0.01s
Presolved: 2126 rows, 2232 columns, 7140 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)
Found heuristic solution: objective 1047065.0000
Found heuristic solution: objective 1045750.0000
Root relaxation: objective 1.003420e+06, 1703 iterations, 0.03 seconds (0.02 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 1003419.59 0 51 1045750.00 1003419.59 4.05% - 0s
H 0 0 1025155.0000 1003419.59 2.12% - 0s
H 0 0 1012530.0000 1003419.59 0.90% - 0s
0 0 1003700.84 0 91 1012530.00 1003700.84 0.87% - 0s
H 0 0 1011550.0000 1003700.84 0.78% - 0s
H 0 0 1011486.0000 1003700.84 0.77% - 0s
0 0 1003702.54 0 113 1011486.00 1003702.54 0.77% - 0s
0 0 1003702.54 0 120 1011486.00 1003702.54 0.77% - 0s
0 0 1003815.55 0 87 1011486.00 1003815.55 0.76% - 0s
0 0 1003906.43 0 59 1011486.00 1003906.43 0.75% - 0s
H 0 0 1008770.0000 1003906.43 0.48% - 0s
0 0 1004239.12 0 121 1008770.00 1004239.12 0.45% - 0s
0 0 1004239.12 0 122 1008770.00 1004239.12 0.45% - 0s
0 0 1004289.66 0 95 1008770.00 1004289.66 0.44% - 0s
0 0 1004289.66 0 111 1008770.00 1004289.66 0.44% - 0s
0 0 1004289.66 0 88 1008770.00 1004289.66 0.44% - 0s
0 0 1004289.66 0 99 1008770.00 1004289.66 0.44% - 0s
0 0 1004289.66 0 72 1008770.00 1004289.66 0.44% - 0s
0 0 1004289.66 0 107 1008770.00 1004289.66 0.44% - 0s
0 0 1004289.66 0 96 1008770.00 1004289.66 0.44% - 0s
0 0 1004289.66 0 75 1008770.00 1004289.66 0.44% - 0s
H 0 0 1006700.0000 1004289.66 0.24% - 0s
0 2 1004289.66 0 75 1006700.00 1004289.66 0.24% - 0s
H 71 77 1006622.0000 1004528.51 0.21% 15.2 0s
H 112 121 1006305.0000 1004528.51 0.18% 12.6 1s
H 170 166 1005559.0000 1004528.51 0.10% 10.8 1s
H 293 290 1005405.0000 1004539.36 0.09% 9.2 1s
* 939 571 38 1005395.0000 1004565.24 0.08% 8.8 1s
H 1196 607 1005350.0000 1004761.71 0.06% 5.0 4s
* 1313 608 45 1005300.0000 1004767.86 0.05% 5.2 4s
3561 974 cutoff 42 1005300.00 1004778.48 0.05% 6.4 5s
24140 4806 1005176.01 40 53 1005300.00 1005008.17 0.03% 8.1 10s
Cutting planes:
Gomory: 44
MIR: 30
Flow cover: 23
Explored 37663 nodes (321687 simplex iterations) in 12.96 seconds (10.68 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 1.0053e+06 1.0053e+06 1.00535e+06 ... 1.00877e+06
Optimal solution found (tolerance 1.00e-04)
Best objective 1.005300000000e+06, best bound 1.005225032424e+06, gap 0.0075%
| $N_t$ | $N_{t-1}$ | $N_t-N_{t-1}$ | $N_{start,t}$ | $N_t-N_{t-1}-N_{start,t}$ | Signification |
|---|---|---|---|---|---|
| 1 | 1 | 0 | 0 | 0 | La centrale fonctionne deux heures consécutives |
| 1 | 0 | 1 | 1 | 0 | la centrale s'allume à l'instant $t$ |
| 0 | 1 | -1 | 0 | -1 | la centrale s'éteint à l'instant $t$ |
| 0 | 0 | 0 | 0 | 0 | La centrale est éteinte deux heures consécutives |
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2785 rows, 2232 columns and 8448 nonzeros
Model fingerprint: 0x460e9919
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
Matrix range [3e-04, 6e+04]
Objective range [1e+00, 2e+03]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 4e+04]
Presolve removed 695 rows and 1269 columns
Presolve time: 0.02s
Presolved: 2090 rows, 963 columns, 6363 nonzeros
Variable types: 672 continuous, 291 integer (291 binary)
Root relaxation: objective 1.011158e+06, 1035 iterations, 0.01 seconds (0.01 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 1011158.36 0 42 - 1011158.36 - - 0s
H 0 0 1024447.0000 1011158.36 1.30% - 0s
0 0 1011766.70 0 37 1024447.00 1011766.70 1.24% - 0s
0 0 1012487.81 0 17 1024447.00 1012487.81 1.17% - 0s
0 0 1012487.81 0 17 1024447.00 1012487.81 1.17% - 0s
H 0 0 1019980.0000 1012487.81 0.73% - 0s
0 0 1012528.31 0 18 1019980.00 1012528.31 0.73% - 0s
0 0 1012528.31 0 18 1019980.00 1012528.31 0.73% - 0s
0 0 1012528.31 0 19 1019980.00 1012528.31 0.73% - 0s
0 0 1012637.32 0 18 1019980.00 1012637.32 0.72% - 0s
0 0 1012637.32 0 19 1019980.00 1012637.32 0.72% - 0s
0 0 1012637.32 0 17 1019980.00 1012637.32 0.72% - 0s
H 0 0 1017238.0000 1012637.32 0.45% - 0s
0 0 1012637.32 0 19 1017238.00 1012637.32 0.45% - 0s
0 0 1012637.32 0 17 1017238.00 1012637.32 0.45% - 0s
0 0 1012637.32 0 18 1017238.00 1012637.32 0.45% - 0s
0 0 1012637.32 0 18 1017238.00 1012637.32 0.45% - 0s
0 0 1012637.32 0 20 1017238.00 1012637.32 0.45% - 0s
0 0 1012637.32 0 20 1017238.00 1012637.32 0.45% - 0s
0 0 1012637.32 0 17 1017238.00 1012637.32 0.45% - 0s
0 2 1012637.32 0 17 1017238.00 1012637.32 0.45% - 0s
H 3 4 1016994.0000 1012637.32 0.43% 9.3 0s
H 27 3 1016430.0000 1015144.08 0.13% 16.1 0s
Cutting planes:
Gomory: 3
MIR: 17
Flow cover: 3
RLT: 7
Explored 49 nodes (2034 simplex iterations) in 0.53 seconds (0.27 work units)
Thread count was 8 (of 8 available processors)
Solution count 6: 1.01643e+06 1.01643e+06 1.01699e+06 ... 1.02445e+06
Optimal solution found (tolerance 1.00e-04)
Best objective 1.016430000000e+06, best bound 1.016409639706e+06, gap 0.0020%
Si la centrale démarre alors on autorise une rampe plus grande. $$ (8.2), P_{t}^{(X,k)}-P_{t-1}^{(X,k)}\leq R_{montée}^{(X)} + N_{start,t}{(X,k)}(R_{démarrage}^{(X)}-R_{montée}^{(X)}) $$
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2785 rows, 2232 columns and 9096 nonzeros
Model fingerprint: 0xe2bdd0ee
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
Matrix range [3e-04, 6e+04]
Objective range [1e+00, 2e+03]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 4e+04]
Presolve removed 24 rows and 0 columns
Presolve time: 0.02s
Presolved: 2761 rows, 2232 columns, 9576 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)
Found heuristic solution: objective 1143740.0000
Root relaxation: objective 9.865633e+05, 1541 iterations, 0.02 seconds (0.02 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 986563.290 0 133 1143740.00 986563.290 13.7% - 0s
H 0 0 1052540.0000 986563.290 6.27% - 0s
H 0 0 999785.00000 986563.290 1.32% - 0s
H 0 0 994685.00000 987113.421 0.76% - 0s
0 0 987113.421 0 126 994685.000 987113.421 0.76% - 0s
0 0 987306.646 0 120 994685.000 987306.646 0.74% - 0s
0 0 987886.769 0 121 994685.000 987886.769 0.68% - 0s
0 0 987887.226 0 122 994685.000 987887.226 0.68% - 0s
0 0 988302.092 0 135 994685.000 988302.092 0.64% - 0s
0 0 988304.690 0 140 994685.000 988304.690 0.64% - 0s
0 0 988309.044 0 152 994685.000 988309.044 0.64% - 0s
0 0 988309.044 0 152 994685.000 988309.044 0.64% - 0s
0 0 988336.401 0 113 994685.000 988336.401 0.64% - 0s
0 0 988343.477 0 123 994685.000 988343.477 0.64% - 0s
0 0 988345.222 0 123 994685.000 988345.222 0.64% - 0s
0 0 988351.317 0 124 994685.000 988351.317 0.64% - 0s
H 0 0 994500.00000 988355.092 0.62% - 0s
0 0 988355.092 0 135 994500.000 988355.092 0.62% - 0s
H 0 0 992340.00000 988355.092 0.40% - 0s
H 0 0 992270.00000 988355.092 0.39% - 0s
0 0 988361.795 0 111 992270.000 988361.795 0.39% - 0s
0 0 988361.907 0 112 992270.000 988361.907 0.39% - 0s
0 0 988374.433 0 125 992270.000 988374.433 0.39% - 0s
0 0 988374.740 0 120 992270.000 988374.740 0.39% - 0s
0 0 988375.536 0 142 992270.000 988375.536 0.39% - 0s
0 0 988375.800 0 146 992270.000 988375.800 0.39% - 0s
0 0 988375.905 0 146 992270.000 988375.905 0.39% - 0s
0 0 988375.905 0 94 992270.000 988375.905 0.39% - 0s
0 2 988375.905 0 94 992270.000 988375.905 0.39% - 0s
H 31 40 990915.00000 988639.353 0.23% 55.5 0s
H 33 40 989677.00000 988639.353 0.10% 52.5 0s
H 77 56 989585.00000 988749.821 0.08% 31.4 0s
H 167 136 989428.41463 988749.821 0.07% 19.3 1s
H 416 341 989425.00000 988899.005 0.05% 12.4 1s
H 1174 597 989350.00000 988930.500 0.04% 12.4 3s
2185 724 infeasible 56 989350.000 989074.814 0.03% 11.5 5s
16213 4019 989349.280 70 7 989350.000 989246.302 0.01% 8.0 10s
Cutting planes:
Gomory: 25
Implied bound: 30
MIR: 57
StrongCG: 1
Flow cover: 41
RLT: 5
Relax-and-lift: 7
Explored 18263 nodes (151362 simplex iterations) in 10.56 seconds (7.70 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 989350 989425 989428 ... 999785
Optimal solution found (tolerance 1.00e-04)
Best objective 9.893500000000e+05, best bound 9.892535739926e+05, gap 0.0097%
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 3433 rows, 2232 columns and 11688 nonzeros
Model fingerprint: 0x7cfd5517
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
Matrix range [3e-04, 6e+04]
Objective range [1e+00, 2e+03]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 4e+04]
Presolve removed 24 rows and 0 columns
Presolve time: 0.02s
Presolved: 3409 rows, 2232 columns, 12168 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)
Found heuristic solution: objective 1179585.0000
Root relaxation: objective 9.870197e+05, 1846 iterations, 0.03 seconds (0.03 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 987019.698 0 136 1179585.00 987019.698 16.3% - 0s
H 0 0 1043335.0000 987019.698 5.40% - 0s
H 0 0 998330.00000 987019.698 1.13% - 0s
H 0 0 993910.00000 987019.698 0.69% - 0s
0 0 987511.972 0 148 993910.000 987511.972 0.64% - 0s
0 0 987547.166 0 149 993910.000 987547.166 0.64% - 0s
0 0 988753.264 0 179 993910.000 988753.264 0.52% - 0s
H 0 0 993735.00000 988753.264 0.50% - 0s
0 0 988753.837 0 166 993735.000 988753.837 0.50% - 0s
H 0 0 993610.00000 988753.837 0.49% - 0s
0 0 989130.923 0 161 993610.000 989130.923 0.45% - 0s
0 0 989192.131 0 159 993610.000 989192.131 0.44% - 0s
0 0 989192.254 0 176 993610.000 989192.254 0.44% - 0s
0 0 989300.395 0 110 993610.000 989300.395 0.43% - 0s
0 0 989303.059 0 135 993610.000 989303.059 0.43% - 0s
0 0 989304.095 0 131 993610.000 989304.095 0.43% - 0s
0 0 989304.095 0 142 993610.000 989304.095 0.43% - 0s
0 0 989305.304 0 158 993610.000 989305.304 0.43% - 0s
0 0 989305.304 0 159 993610.000 989305.304 0.43% - 0s
0 0 989306.359 0 192 993610.000 989306.359 0.43% - 0s
0 0 989306.359 0 149 993610.000 989306.359 0.43% - 0s
0 2 989306.359 0 126 993610.000 989306.359 0.43% - 0s
H 78 74 990948.00000 989835.717 0.11% 34.6 1s
H 169 148 990935.00000 989835.717 0.11% 25.0 1s
* 311 232 63 990915.00000 989835.717 0.11% 20.5 1s
1406 706 cutoff 23 990915.000 990039.934 0.09% 21.5 5s
H 1866 767 990870.00000 990039.934 0.08% 20.9 6s
* 7187 3575 77 990865.00000 990267.120 0.06% 16.0 8s
9433 5168 990704.032 64 14 990865.000 990279.323 0.06% 15.5 10s
*12001 6476 75 990860.00000 990292.232 0.06% 15.5 11s
H14446 6401 990815.00000 990299.913 0.05% 15.8 12s
14474 6418 990341.946 26 43 990815.000 990299.913 0.05% 16.0 15s
H14687 6137 990805.00000 990299.913 0.05% 16.1 16s
23593 8705 cutoff 71 990805.000 990384.938 0.04% 15.4 20s
33190 10204 990792.699 79 49 990805.000 990411.485 0.04% 15.5 25s
44436 15055 cutoff 54 990805.000 990433.200 0.04% 15.4 36s
H46813 14752 990780.00000 990436.783 0.03% 15.4 37s
52174 16749 990676.457 57 99 990780.000 990444.650 0.03% 15.6 40s
61827 19680 990594.126 60 43 990780.000 990461.111 0.03% 15.9 45s
71587 22182 cutoff 52 990780.000 990475.384 0.03% 16.0 50s
79962 24156 990585.926 56 15 990780.000 990488.219 0.03% 16.1 55s
89908 26143 990748.442 62 62 990780.000 990502.353 0.03% 16.3 60s
98157 27567 990736.227 65 66 990780.000 990513.075 0.03% 16.3 65s
106638 28799 cutoff 57 990780.000 990522.795 0.03% 16.3 70s
116161 30088 990772.261 57 27 990780.000 990532.185 0.03% 16.3 75s
126851 30824 cutoff 59 990780.000 990542.614 0.02% 16.3 80s
138095 31552 990768.885 62 56 990780.000 990551.026 0.02% 16.3 85s
148720 31934 cutoff 54 990780.000 990560.462 0.02% 16.3 90s
161061 31863 cutoff 60 990780.000 990570.756 0.02% 16.2 95s
172015 31603 cutoff 51 990780.000 990581.090 0.02% 16.2 100s
183708 31261 cutoff 56 990780.000 990591.853 0.02% 16.1 105s
196225 30377 990666.863 61 81 990780.000 990604.258 0.02% 16.0 110s
206360 30330 990655.654 61 23 990780.000 990611.544 0.02% 15.9 115s
218079 29567 cutoff 61 990780.000 990624.485 0.02% 15.8 120s
230692 28964 990730.856 58 86 990780.000 990638.115 0.01% 15.7 125s
244695 28429 990654.245 63 20 990780.000 990653.874 0.01% 15.5 130s
255977 28532 990697.745 58 18 990780.000 990661.512 0.01% 15.4 135s
269013 28953 990738.571 61 4 990780.000 990671.249 0.01% 15.3 140s
281508 29021 cutoff 56 990780.000 990680.419 0.01% 15.1 145s
Cutting planes:
Gomory: 15
Implied bound: 82
Projected implied bound: 6
MIR: 289
StrongCG: 6
Flow cover: 71
RLT: 3
Relax-and-lift: 4
Explored 282970 nodes (4283677 simplex iterations) in 145.53 seconds (165.09 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 990780 990780 990805 ... 990948
Optimal solution found (tolerance 1.00e-04)
Best objective 9.907800000000e+05, best bound 9.906816144175e+05, gap 0.0099%
On a la table suivante : |$N_t$|$N_{t-1}$|$N_t-N_{t-1}$|$N_{start,t}$|$N_t-N_{t-1}-N_{start,t}$|Signification| |---|---|---|---|---|---| |1|1|0|0|0|La centrale fonctionne deux heures consécutives| |1|0|1|1|0|la centrale s'allume à l'instant $t$| |0|1|-1|0|-1|la centrale s'éteint à l'instant $t$| |0|0|0|0|0|La centrale est éteinte deux heures consécutives| $$ \forall t,\forall t' \in \{1,\dots,8\}, \forall k, N_{start,t}^{(X,k)}\leq N_{t+t'}^{(X,k)} , \text{La centrale fonctionne au moins 8 heures après le démarrage} $$
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 7105 rows, 2232 columns and 17088 nonzeros
Model fingerprint: 0xfb554425
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
Matrix range [3e-04, 6e+04]
Objective range [1e+00, 2e+03]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 4e+04]
Presolve removed 645 rows and 0 columns
Presolve time: 0.04s
Presolved: 6460 rows, 2232 columns, 15678 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)
Found heuristic solution: objective 1028375.0000
Root relaxation: objective 9.851845e+05, 1596 iterations, 0.03 seconds (0.04 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 985184.550 0 63 1028375.00 985184.550 4.20% - 0s
H 0 0 990215.00000 985184.550 0.51% - 0s
H 0 0 989865.00000 985184.550 0.47% - 0s
H 0 0 989790.00000 985761.648 0.41% - 0s
0 0 985761.648 0 110 989790.000 985761.648 0.41% - 0s
0 0 985872.918 0 77 989790.000 985872.918 0.40% - 0s
0 0 986085.714 0 49 989790.000 986085.714 0.37% - 0s
0 0 986150.000 0 62 989790.000 986150.000 0.37% - 0s
0 0 986449.728 0 61 989790.000 986449.728 0.34% - 0s
0 0 986528.505 0 65 989790.000 986528.505 0.33% - 0s
0 0 986528.505 0 65 989790.000 986528.505 0.33% - 0s
0 0 986528.505 0 65 989790.000 986528.505 0.33% - 0s
0 0 986528.505 0 63 989790.000 986528.505 0.33% - 0s
H 0 0 989615.00000 986528.505 0.31% - 0s
0 2 986528.505 0 63 989615.000 986528.505 0.31% - 0s
H 107 104 989283.00000 986564.240 0.27% 21.7 0s
H 115 104 988945.00000 986564.240 0.24% 21.0 0s
H 150 141 988490.00000 986564.240 0.19% 19.6 0s
H 204 151 987855.00000 986566.871 0.13% 18.0 1s
H 961 351 987777.00000 986839.834 0.09% 13.7 1s
H 1177 446 987719.00000 986862.410 0.09% 13.0 1s
3236 940 987649.879 28 10 987719.000 987122.302 0.06% 12.8 5s
Cutting planes:
Gomory: 1
Implied bound: 1
Projected implied bound: 6
Clique: 6
MIR: 18
StrongCG: 3
Flow cover: 20
Zero half: 1
Relax-and-lift: 4
Explored 7237 nodes (77546 simplex iterations) in 6.25 seconds (5.85 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 987719 987777 987855 ... 990215
Optimal solution found (tolerance 1.00e-04)
Best objective 9.877190000000e+05, best bound 9.876657333333e+05, gap 0.0054%
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 11321 rows, 2232 columns and 33556 nonzeros
Model fingerprint: 0xbd76973d
Variable types: 672 continuous, 1560 integer (1560 binary)
Coefficient statistics:
Matrix range [3e-04, 6e+04]
Objective range [1e+00, 2e+03]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 4e+04]
Presolve removed 3926 rows and 0 columns
Presolve time: 0.13s
Presolved: 7395 rows, 2232 columns, 32998 nonzeros
Variable types: 672 continuous, 1560 integer (1560 binary)
Found heuristic solution: objective 1291446.0000
Found heuristic solution: objective 1121845.0000
Found heuristic solution: objective 1005080.0000
Root relaxation: objective 9.859333e+05, 1924 iterations, 0.04 seconds (0.04 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 985933.334 0 52 1005080.00 985933.334 1.90% - 0s
H 0 0 999470.00000 985933.334 1.35% - 0s
0 0 986537.818 0 131 999470.000 986537.818 1.29% - 0s
0 0 986578.142 0 125 999470.000 986578.142 1.29% - 0s
H 0 0 992540.00000 986578.142 0.60% - 0s
0 0 986806.992 0 115 992540.000 986806.992 0.58% - 0s
0 0 986806.992 0 127 992540.000 986806.992 0.58% - 0s
0 0 986893.560 0 185 992540.000 986893.560 0.57% - 0s
0 0 987023.402 0 171 992540.000 987023.402 0.56% - 0s
0 0 987029.760 0 179 992540.000 987029.760 0.56% - 0s
0 0 987034.496 0 212 992540.000 987034.496 0.55% - 0s
H 0 0 991037.00000 987098.675 0.40% - 0s
0 0 987098.675 0 192 991037.000 987098.675 0.40% - 0s
0 0 987141.967 0 130 991037.000 987141.967 0.39% - 0s
0 0 987141.967 0 133 991037.000 987141.967 0.39% - 0s
0 0 987329.470 0 155 991037.000 987329.470 0.37% - 0s
0 0 987332.695 0 208 991037.000 987332.695 0.37% - 0s
0 0 987342.806 0 184 991037.000 987342.806 0.37% - 1s
0 0 987344.080 0 209 991037.000 987344.080 0.37% - 1s
0 0 987345.432 0 217 991037.000 987345.432 0.37% - 1s
0 0 987372.112 0 196 991037.000 987372.112 0.37% - 1s
0 0 987383.806 0 226 991037.000 987383.806 0.37% - 1s
0 0 987385.942 0 221 991037.000 987385.942 0.37% - 1s
0 0 987386.382 0 227 991037.000 987386.382 0.37% - 1s
0 0 987386.654 0 213 991037.000 987386.654 0.37% - 1s
0 0 987407.915 0 231 991037.000 987407.915 0.37% - 1s
0 0 987411.926 0 231 991037.000 987411.926 0.37% - 1s
0 0 987415.396 0 195 991037.000 987415.396 0.37% - 1s
0 0 987415.396 0 185 991037.000 987415.396 0.37% - 1s
0 0 987453.956 0 211 991037.000 987453.956 0.36% - 1s
0 0 987453.956 0 217 991037.000 987453.956 0.36% - 1s
0 0 987456.190 0 212 991037.000 987456.190 0.36% - 1s
0 0 987456.190 0 217 991037.000 987456.190 0.36% - 1s
0 0 987460.875 0 233 991037.000 987460.875 0.36% - 1s
0 0 987461.142 0 172 991037.000 987461.142 0.36% - 1s
0 0 987461.228 0 186 991037.000 987461.228 0.36% - 1s
0 0 987467.132 0 195 991037.000 987467.132 0.36% - 1s
0 0 987467.232 0 192 991037.000 987467.232 0.36% - 1s
0 0 987478.868 0 157 991037.000 987478.868 0.36% - 1s
0 0 987478.868 0 172 991037.000 987478.868 0.36% - 1s
0 0 987479.993 0 179 991037.000 987479.993 0.36% - 1s
0 0 987480.568 0 162 991037.000 987480.568 0.36% - 1s
0 0 987480.568 0 154 991037.000 987480.568 0.36% - 1s
0 0 987481.390 0 173 991037.000 987481.390 0.36% - 1s
0 0 987481.390 0 173 991037.000 987481.390 0.36% - 1s
H 0 0 990865.00000 987481.390 0.34% - 2s
H 0 0 990605.00000 987481.390 0.32% - 2s
0 2 987481.390 0 173 990605.000 987481.390 0.32% - 2s
H 135 109 989610.00000 987730.327 0.19% 104 3s
H 137 109 989100.00000 987730.327 0.14% 102 3s
H 174 119 989090.00000 987733.265 0.14% 89.0 3s
H 193 123 989085.00000 987733.265 0.14% 82.4 3s
H 227 132 989052.00000 987733.265 0.13% 76.4 3s
830 401 988604.427 27 64 989052.000 987765.437 0.13% 45.4 5s
H 1151 541 989005.00000 987768.630 0.13% 39.2 7s
1170 556 987768.630 13 229 989005.000 987768.630 0.13% 42.4 10s
H 1305 580 988753.00000 988072.404 0.07% 41.9 10s
5132 1125 988445.929 38 39 988753.000 988443.992 0.03% 27.5 15s
9407 1247 cutoff 45 988753.000 988516.114 0.02% 25.5 20s
Cutting planes:
Gomory: 3
Implied bound: 1
Clique: 1
MIR: 171
Flow cover: 53
Zero half: 11
RLT: 1
Explored 12900 nodes (312643 simplex iterations) in 22.31 seconds (22.78 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 988753 989005 989052 ... 991037
Optimal solution found (tolerance 1.00e-04)
Best objective 9.887530000000e+05, best bound 9.886580344989e+05, gap 0.0096%